Water layer in SE is a transparent sphere with a blue color (or green and red now), intersecting with the surface at a certain height. If you mean exporting color texture with oceans filled with blue, it requires additional coding.

When I export the textures, I open the surface color map in Paint.NET and get a really see-through blank canvas, I have to use the eyedrop tool to see the colors there. How do I turn it into a map of the planet ???

I open the surface color map in Paint.NET and get a really see-through blank canvas, I have to use the eyedrop tool to see the colors there. How do I turn it into a map of the planet ???

You really should use a more advanced image editing program like GIMP so that you have the ability to create and manipulate layer masks. Surfaces are transparent, or partially so, so that they can carry specular reflection data in the alpha channel (i.e. the alpha channel is a specular map). The best way to use these maps would be to use GIMP and to create a layer mask for that map, and transfer the layer's alpha channel to the mask. Then you can toggle the mask as you like to see the surface, delete the mask altogether to eliminate the alpha channel, or even extract it into a new layer to save as a separate map.

Can I export textures of just an area of the planet/moon/asteroid/comet like an island or a particularly pretty area?

No, you can only export the entire world's texture. SpaceEngineer has thought about implementing a way of exporting a small area in high res, but that would not be implemented until some time in the future.